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Pigeonhole Principle Worksheet for Maths Olympiad

Maths Olympiad topic 23: Pigeonhole and worst case · Set 60 · 40 questions
TalentJR

How to solve it (Think of the unluckiest case): Imagine the unluckiest draws: how many can you take and still NOT have what you want? Add 1 to that number — the next one must give it. Pigeonhole: with n boxes (colours, months, days), n × (k − 1) + 1 things make sure some box has at least k.

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  1. 1.A drawer has lots of white, black and red socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 3 (b) 6 (c) 2 (d) 4
  2. 2.A drawer has lots of red, green, black, white and blue socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 5 (b) 10 (c) 2 (d) 6
  3. 3.A drawer has lots of black, white, green, blue and red socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 10 (c) 6 (d) 5
  4. 4.A bag has 7 red balls and 8 white balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 8 (b) 2 (c) 9 (d) 15
  5. 5.What is the smallest number of children in a group that makes sure at least three of them were born in the same day of the week? (a) 22 (b) 15 (c) 21 (d) 14
  6. 6.A bag has 5 red balls and 3 blue balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 4 (b) 2 (c) 8 (d) 6
  7. 7.A bag has 12 blue balls and 5 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 17 (b) 13 (c) 14 (d) 2
  8. 8.A bag has 11 red balls and 6 blue balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 12 (c) 17 (d) 7
  9. 9.A bag has 5 blue balls and 8 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 6 (b) 13 (c) 2 (d) 7
  10. 10.A bag has 9 green balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 11 (b) 13 (c) 2 (d) 10
  11. 11.A drawer has lots of yellow, red, black and green socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 4 (b) 5 (c) 8 (d) 2
  12. 12.A bag has 10 black balls and 5 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 15 (b) 11 (c) 2 (d) 12
  13. 13.A bag has 8 white balls and 5 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 9 (b) 13 (c) 2 (d) 10
  14. 14.What is the smallest number of children in a group that makes sure at least two of them were born in the same month? (a) 24 (b) 12 (c) 13 (d) 25
  15. 15.A bag has 4 blue balls and 5 green balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 5 (b) 9 (c) 2 (d) 6
  16. 16.What is the smallest number of children in a group that makes sure at least three of them were born in the same month? (a) 37 (b) 24 (c) 25 (d) 36
  17. 17.A drawer has lots of yellow, white, red and black socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 8 (b) 4 (c) 5 (d) 2
  18. 18.A bag has 7 black balls and 9 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 2 (b) 16 (c) 9 (d) 8
  19. 19.A bag has 3 blue balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 15 (b) 4 (c) 2 (d) 5
  20. 20.A drawer has lots of black, green, white and red socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 4 (b) 5 (c) 2 (d) 8
  21. 21.A drawer has lots of blue, yellow, white, green and red socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 10 (b) 2 (c) 6 (d) 5
  22. 22.A drawer has lots of yellow, green and black socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 2 (b) 6 (c) 4 (d) 3
  23. 23.A bag has 7 red balls and 3 black balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 4 (b) 10 (c) 2 (d) 8
  24. 24.A bag has 5 black balls and 6 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 11 (b) 6 (c) 2 (d) 7
  25. 25.A bag has 9 green balls and 11 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 11 (b) 2 (c) 10 (d) 20
  26. 26.A bag has 12 blue balls and 4 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 2 (b) 5 (c) 13 (d) 16
  27. 27.A drawer has lots of black, red, yellow and blue socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 5 (b) 2 (c) 4 (d) 8
  28. 28.A drawer has lots of red and white socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 4 (b) 5 (c) 2 (d) 3
  29. 29.What is the smallest number of children in a group that makes sure at least two of them were born in the same day of the week? (a) 8 (b) 15 (c) 7 (d) 14
  30. 30.A bag has 10 green balls and 7 blue balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 11 (b) 17 (c) 8 (d) 2
  31. 31.A bag has 3 blue balls and 3 red balls. Without looking, how many balls must you take out to be sure of getting 2 red balls? (a) 5 (b) 2 (c) 4 (d) 6
  32. 32.A bag has 12 red balls and 11 yellow balls. Without looking, how many balls must you take out to be sure of getting 2 yellow balls? (a) 14 (b) 2 (c) 13 (d) 23
  33. 33.A bag has 6 yellow balls and 4 blue balls. Without looking, how many balls must you take out to be sure of getting 2 blue balls? (a) 8 (b) 10 (c) 2 (d) 7
  34. 34.A bag has 10 yellow balls and 12 red balls. Without looking, how many balls must you take out to be sure of getting at least one ball of each colour? (a) 22 (b) 11 (c) 13 (d) 2
  35. 35.A bag has 5 yellow balls and 12 white balls. Without looking, how many balls must you take out to be sure of getting 2 white balls? (a) 6 (b) 2 (c) 7 (d) 17
  36. 36.A drawer has lots of green and yellow socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 4 (b) 5 (c) 3 (d) 2
  37. 37.A drawer has lots of blue, red, green and black socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? (a) 5 (b) 2 (c) 8 (d) 4
  38. 38.What is the smallest number of children in a group that makes sure at least four of them were born in the same month? (a) 49 (b) 48 (c) 36 (d) 37
  39. 39.A bag has 9 red balls and 10 green balls. Without looking, how many balls must you take out to be sure of getting 2 green balls? (a) 19 (b) 10 (c) 11 (d) 2
  40. 40.What is the smallest number of children in a group that makes sure at least four of them were born in the same day of the week? (a) 29 (b) 22 (c) 21 (d) 28

Answer key

  1. 4
  2. 6
  3. 6
  4. 9
  5. 15
  6. 6
  7. 14
  8. 12
  9. 7
  10. 11
  11. 5
  12. 12
  13. 10
  14. 13
  15. 6
  16. 25
  17. 5
  18. 9
  19. 5
  20. 5
  21. 6
  22. 4
  23. 8
  24. 7
  25. 11
  26. 13
  27. 5
  28. 3
  29. 8
  30. 11
  31. 5
  32. 14
  33. 8
  34. 13
  35. 7
  36. 3
  37. 5
  38. 37
  39. 11
  40. 22

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